Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

18 paolo mancosuamount of humility is needed to avoid the risk of theorizing without keepingour feet on the ground. It is interesting to note that also a recent volumewritten by historians and philosophers of mathematics (Ferreiros and Gray 2006)displays the same modesty with respect to these metaphilosophical issues. Atthe same time, the number of topics we touch upon is immeasurably vasterthan the ones addressed by the Lakatos tradition. Visualization, diagrammaticreasoning, purity of methods, category theory, mathematical physics, and manyother topics we investigate here are remarkably absent from a tradition whichhas made attention to mathematical practice its call to arms. One exceptionhere is Corfield (2003), which does indeed touch upon many of the topics westudy. However, and this is another important point, we differ from Corfieldin two essential points. First of all, the authors of this collection do not engagein polemic with the foundationalist tradition and, as a matter of fact, many ofthem work, or have worked, also as mathematical logicians (of course, thereare differences of attitude, vis-à-vis foundations, among the contributors). Weare, by and large, calling for an extension to a philosophy of mathematics thatwill be able to address topics that the foundationalist tradition has ignored. Butthat does not mean that we think that the achievements of this tradition shouldbe discarded or ignored as being irrelevant to philosophy of mathematics.Second, unlike Corfield, we do not dismiss the analytic tradition in philosophyof mathematics but rather seek to extend its tools to a variety of areas that havebeen, by and large, ignored. For instance, to give one example among many,the chapter on explanation shows how the topic of mathematical explanation isconnected to two major areas of analytic philosophy: indispensability argumentsand models of scientific explanation. But this conciliatory note should not hidethe force of our message: we think that the aspects of mathematical practicewe investigate are absolutely vital to an understanding of mathematics andthat having ignored them has drastically impoverished analytic philosophy ofmathematics.In the Lakatos tradition, it was Kitcher in particular who attempted to builda bridge with analytic philosophy. For instance, when engaged in his workon explanation in philosophy of science he also made sure that mathematicalexplanation was also taken into account. We are less ambitious than Kitcher, inthat we do not propose a unified epistemology and ontology of mathematicsand a theory of how mathematical knowledge grows rationally. But we aremuch more ambitious in another respect, in that we cover a broad spectrumof case studies arising from mathematical practice which we subject to analyticinvestigation. Thus, in addition to the case of explanation already mentioned,Giaquinto investigates whether synthetic a priori knowledge can be obtainedby appealing to experiences of visualization; Tappenden engages recent work